We consider the decomposition of a compact-type symmetric space into a product of factors and show that the rank-one factors, when considered as totally geodesic submanifolds of the space, are isolated from inequivalent minimal submanifolds.

We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curva...

Warped product CR-submanifolds in Kählerian manifolds were intensively studied only since 2001 after the impulse given by B.Y. Chen in [2], [3]. Immediately after, another line of research, similar to that concerning Sasakian geometry as the odd dimensional version of Kählerian geometry, was developed, namely warped product contact CR-submanifolds in Sasakian manifolds (cf. [6], [7]). In this n...

Recently, we researched certain twisted product CR-submanifolds in a Kaehler manifold and some inequalities of the second fundamental form of this submanifold ([11]). In this talk, we consider two kind twisted product CR-submanifolds in a locally conformal Kaehler manifold. In these submanifolds, we give a inequality of the second fundamental form (See Theorems 5.3 and 5.4) and consider the equ...

The main objective of this paper is to discuss a convexity theorem for a certain class of Riemannian manifolds, so-called isoparametric submanifolds, and how this relates to other convexity theorems. In the introduction we will present the convexity theorems. In Section 2 we will describe the geometry of isoparametric submanifolds and in Section 3 we will relate this to the geometries of the ot...

We study the problem of counting instantons with coassociative boundary condition in (almost) G2-manifolds. This is analog to the open GromovWitten theory for counting holomorphic curves with Lagrangian boundary condition in Calabi-Yau manifolds. We explain its relationship with the Seiberg-Witten invariants for coassociative submanifolds. Intersection theory of Lagrangian submanifolds is an es...

In this paper we generalize the monotonicity formulas of [C] for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., [A], [CM1] and [GL] for applications of monotonicity to uniqueness. Among the applications here is that level sets of Green’s function on open manifolds with nonnegative Ricci curvature are asymptotically umbilic.